How to Calculate a Phi Coefficient in R

In this article, we will discuss what is Phi Coefficient and How to Calculate a Phi Coefficient in R Programming Language.

What is the Phi Coefficient?

The Phi coefficient, also known as the Phi correlation coefficient or the coefficient of association, is a measure of association between two binary variables. It is similar to Pearson’s correlation coefficient but is specifically used for categorical data arranged in a 2×2 contingency table.

The Phi coefficient ranges from -1 to 1:

• If the value is close to 1, it indicates a strong positive association between the two variables (i.e., as one variable increases, the other tends to increase).
• If the value is close to -1, it indicates a strong negative association between the two variables (i.e., as one variable increases, the other tends to decrease).
• If the value is close to 0, it indicates no association between the two variables.

Formula:

The formula to compute the Phi coefficient for a 2×2 contingency table is:

[Tex]\phi = \frac{(ad – bc)}{\sqrt{(a + b)(c + d)(a + c)(b + d)}} [/Tex]

Where:

𝑎, 𝑏, 𝑐 and 𝑑 are the frequencies of the four cells in the contingency table.

We have collected data on the smoking habits and lung cancer incidence among a sample of individuals. We want to investigate the association between smoking status (smoker or non-smoker) and lung cancer (yes or no). We construct a 2×2 contingency table to summarize the data:

Category	Lung Cancer	No Lung Cancer
Smoker	30	20
Non-Smoker	10	40

• 𝑎 = 30 represents the number of smokers who have lung cancer.
• 𝑏 = 20 represents the number of smokers who do not have lung cancer.
• 𝑐 = 10 represents the number of non-smokers who have lung cancer.
• 𝑑 = 40 represents the number of non-smokers who do not have lung cancer.

Implementation of Formula:

[Tex]\phi = \frac{(30 \times 40 – 20 \times 10)}{\sqrt{(30 + 20)(10 + 40)(30 + 10)(20 + 40)}} [/Tex]

= (1200-200)/√(50)(50)(40)(60)

= 1000/√6000000

≈ 100/2449

≈ 0.408

A Phi coefficient of 0.408 indicates a moderate positive association between smoking status and lung cancer. This means that smokers are more likely to have lung cancer compared to non-smokers, but the association is not extremely strong.

The Phi coefficient ranges from -1 to 1:

• If the value is close to 1, it indicates a strong positive association between the two variables.
• If the value is close to -1, it indicates a strong negative association between the two variables.
• If the value is close to 0, it indicates no association between the two variables.

So, a Phi coefficient of 0.408 suggests that there is a moderate positive association between smoking status and lung cancer in the sample.

Phi Coefficient in R

Calculating a Phi coefficient in R can be done using the assocstats() function from the vcd package.

First, install and load the vcd package.

R

install.packages("vcd") library(vcd)

Then use the assocstats() function to compute various association statistics including the Phi coefficient for a 2×2 contingency table.

R

# Create a 2x2 contingency table data <- matrix(c(20, 30, 10, 40), nrow = 2) # Compute association statistics result <- assocstats(data) # Print the Phi coefficient print(result$phi)

Output:

[1] 0.2182179

Calculate a Phi Coefficient in R using psych

We have collected data on the relationship between exercise habits (regular exercise or no regular exercise) and heart disease (yes or no) among a sample of individuals.

we have the following contingency table:

Category	Heart Disease	No Heart Disease
Regular Exercise	50	20
No Regular Exercise	30	40

We want to calculate the Phi coefficient to determine the association between exercise habits and heart disease.

R

# Load the psych package library(psych) # Create the contingency table data <- matrix(c(50, 20, 30, 40), nrow = 2) # Compute association statistics phi(data, digits = 4)

Output:

[1] 0.2887

psych is used for various statistical analyses, including the computation of association statistics like the phi coefficient.

• We creates a 2×2 contingency table with the specified values. It might represent the frequency of occurrences across two categorical variables.
• phi function calculates the phi coefficient for the given 2×2 contingency table. The phi coefficient measures the strength of association between two categorical variables in a 2×2 table.
• The digits = 4 parameter specifies that the result should be rounded to four decimal places.

The output represents the Phi coefficient calculated for the given contingency table. It indicates a moderate positive association between exercise habits and heart disease in the sample.

Uses of Phi Coefficient

The Phi coefficient has several uses in statistical analysis, particularly in the categorical data and association between binary variables.

1. Measuring Association: It quantifies the degree of association between two binary variables in a 2×2 contingency table.
2. Determining Strength of Association: Phi coefficient values close to 1 or -1 indicate a strong association, while values close to 0 suggest a weak association.
3. Comparing Association between Variables: It allows for comparison of the strength of association between different pairs of binary variables.
4. Hypothesis Testing: It can be used in hypothesis testing to determine whether the observed association between variables is statistically significant.
5. Variable Selection: In exploratory data analysis, Phi coefficient can help in selecting variables for further analysis or modeling based on their association with the outcome variable.
6. Epidemiological Studies: In epidemiology, Phi coefficient is used to assess the association between risk factors and disease outcomes in observational studies.
7. Social Sciences Research: It is used in social sciences research to analyze relationships between categorical variables such as gender, ethnicity, and voting behavior.

Conclusion

In conclusion, calculating a Phi coefficient in R provides a straightforward and efficient method for quantifying the association between two binary variables. Utilizing the assocstats() function from the vcd package, researchers can quickly obtain Phi coefficients to assess the strength and significance of relationships in categorical data. This statistical measure offers valuable insights into various fields, including epidemiology, social sciences, and market research, enabling informed decision-making and further exploration of associations between variables.